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CRITICISMS AND DISCUSSIONS. 309 

To bring these magical squares to the surface the squares of 
each set of parallel squares may be permuted as follows: 

Original order 1, 2, 3, 4, 5, 6, 

Permuted order 3, 2, 1, 6, 5, 4. 

The result is the final cube shown in the beginning of this article. 

The above permutation is subject to two conditions. The sev- 
eral sets of parallel squares must all be permuted in the same man- 
ner. Any two parallel squares which in the original cube are located 
on opposite sides of the middle plane of the cube and at an equal 
distance from it, in the permuted cube must be located on opposite 
sides of the middle plane of the cube and at an equal distance from it. 
These conditions are for the protection of the diagonals. 

John Worthington. 

MAGIC IN THE FOURTH DIMENSION. 

Definition of terms : Row is a general term ; rank denotes a hori- 
zontal right-to-left row; file a row from front to back; and column 
a vertical row in a cube — not used of any horizontal dimension. 

If n 2 numbers of a given series can be grouped so as to form a 

magic square and n such squares be so placed as to constitute a 

magic cube, why may we not go a step further and group n cubes 

in relations of the fourth dimension ? In a magic square containing 

the natural series 1 . . . w 2 the summation is— — ' ; in a magic cube 

fl( fl^-i— T ^ 

with the series i...» 3 it is — - — ^— -; and in an analogous fourth- 

dimension construction it naturally will be — - — — — . 

With this idea in mind I have made some experiments, and the 
results are interesting. The analogy with squares and cubes is not 
perfect, for rows of numbers can be arranged side by side to repre- 
sent a visible square, squares can be piled one upon another to make 
a visible cube, but cubes cannot be so combined in drawing as to 
picture to the eye their higher relations. My expectation a priori 
was that some connection or relation, probably through some form 
of diagonal-of-diagonal, would be found to exist between the cubes 
containing the n* terms of a series. This particular feature did ap- 
pear in the cases where n was odd. Here is how it worked out: 

I. When n is odd. 
1. Let w=3, then 8=123. — The natural series 1. . .81 was di- 
vided into three sub-series such that the sum of each would be 



3io 



THE MONIST. 



one-third the sum of the whole. In dealing with any such series 
when n is odd there will be n sub-series, each starting with one of 
the first w numbers, and the difference between successive terms will 
be w+i, except after a multiple of n, when the difference is i. In 
the present case the three sub-series begin respectively with i, 2, 3, 
and the first is 1 5 9 10 14 18 19 23 27 28 32 36 37 41 45 46 50 54 
55 59 63 64 68 72 73 yy 81. These numbers were arranged in 
three squares constituting a magic cube, and the row of squares 
so formed was flanked on right and left by similar rows formed from 
the other two sub-series (see Fig. 1). 

It is not easy — perhaps it is not possible — to make an abso- 
lutely perfect cube of 3. These are not perfect, yet they have many 



25 


38 


60 


28 


77 


18 


67 


8 


48 


33 


79 


11 


72 


1 


SO 


21 


40 


62 


65 


6 


52 


23 


45 


55 


35 


75 


13 


29 


78 


16 


68 


9 


46 


26 


39 


58 


70 


2 


5i 


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63 


31 


80 


12 


24 


43 


56 


36 


73 


14 


66 


4 


53 


69 


7 


47 


27 


37 


59 


30 


76 


17 


20 


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61 


32 


81 


10 


71 


3 


49 


34 


74 


15 


64 


5 


54 


22 


44 


57 



Fig. 1. (3') 

striking features. Taking the three cubes separately we find that 
in each all the "straight" dimensions — rank, file and column — have 
the proper footing, 123. In the middle cube there are two plane 
diagonals having the same summation, and in cubes I and III one 
each. In cube II four cubic diagonals and four diagonals of vertical 
squares are correct ; I and III each have one cubic diagonal and one 
vertical-square diagonal. 

So much for the original cubes; now for some combinations. 
The three squares on the diagonal running down from left to right 
will make a magic cube with rank, file, column, cubic diagonals, 
two plane diagonals and four vertical-square diagonals (37 in all) 
correct. Two other cubes can be formed by starting with the top 
squares of II and III respectively and following the "broken diag- 



CRITICISMS AND DISCUSSIONS. 



3" 



onals" running downward to the right. In each of these S occurs 
at least 28 times (in 9 ranks, 9 files, 9 columns and one cubic diag- 
onal). Various other combinations may be found by taking the 
squares together in horizontal rows and noting how some columns 
and assorted diagonals have the proper summation, but the most 
important and significant are those already pointed out. In all the 
sum 123 occurs over 200 times in this small figure. 



317 


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223 


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286 


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323 


454 


485 


5" 


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198 


329 


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386 


542 


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204 


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573 


79 


423 


554 


85 


236 


267 


298 


429 


585 


ITI 


1+2 


'73 


304 


460 


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48 


179 


335 


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517 


548 


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392 


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367 


398 


529 


60 


86 


242 


273 


404 


560 


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435 


461 


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23 


154 


3'0 


336 


492 


523 


29 


185 


504 


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186 


342 


498 


379 


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217 


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424 


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237 


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405 


556 


587 


118 


149 


280 


43' 


462 


618 


24 


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306 


337 


493 


524 


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212 


368 


399 


530 


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380 


531 


62 


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562 


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130 


281 


437 


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624 


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499 


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176 


332 


488 


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550 


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425 


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269 


300 


426 


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589 


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608 


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30S 



Fig. 2. (s 4 ) 



One most interesting fact remains to be noticed. While the 
three cubes were constructed separately and independently the figure 
formed by combining them is an absolutely perfect square of 9, with 
a summation of 369 in rank, file and corner diagonal (besides all 
"broken" diagonals running downward to the right), and a perfect 



312 THE MONIST. 

balancing of complementary numbers about the center. Any such 
pair, taken with the central number 41, gives us the familiar sum 123, 
and this serves to bind the whole together in a remarkable manner. 

2. Let w=5, then 8=1565. — In Fig. 2 is represented a group 
of 5-cubes each made up of the numbers in a sub-series of the nat- 
ural series 1 . . .625. In accordance with the principle stated in a pre- 
vious paragraph the central sub-series is 1 7 13 19 25 26 32 ... 625, 
and the other four can easily be discovered by inspection. Each of 
the twenty-five small squares has the summation 1565 in rank, file, 
corner diagonal and broken diagonals, twenty times altogether in 
each square, or 500 times for all. 

Combining the five squares in col. I we have a cube in which 
all the 75 "straight" rows (rank, file and vertical column), all the 
horizontal diagonals and three of the four cubic diagonals foot up 
1565. In cube III all the cubic diagonals are correct. Each cube 
also has seven vertical-square diagonals with the same summation. 
Taking together the squares in horizontal rows we find certain 
diagonals having the same sum, but the columns do not. The five 
squares in either diagonal of the large square, however, combine to 
produce almost perfect cubes, with rank, file, column and cubic 
diagonals all correct, and many diagonals of vertical squares. 

A still more remarkable fact is that the squares in the broken 
diagonals running in either direction also combine to produce cubes 
as nearly perfect as those first considered. Indeed, the great square 
seems to be an enlarged copy of the small squares, and where the 
cells in the small ones unite to produce S the corresponding squares 
in the large figure unite to produce cubes more or less perfect. 
Many other combinations are discoverable, but these are sufficient 
to illustrate the principle, and show the interrelations of the cubes 
and their constituent squares. The summation 1565 occurs in this 
figure not less than 1400 times. 

The plane figure containing the five cubes (or twenty-five 
squares) is itself a perfect square with a summation of 7825 
for every rank, file, corner or broken diagonal. Furthermore all 
complementary pairs are balanced about the center, as in Fig. 1. 
Any square group of four, nine or sixteen of the small squares is 
magic, and if the group of nine is taken at the center it is "perfect." 
It is worthy of notice that all the powers of « above the first lie in 
the middle rank of squares, and that all other multiples of « are 
grouped in regular relations in the other ranks and have the same 



CRITICISMS AND DISCUSSIONS. 



313 



grouping in all the squares of any given rank. The same is true 
of the figure illustrating 7*, which is to be considered next. 

3. Let n=7, then 8=8407. — This is so similar in all its prop- 
erties to the 5-construction just discussed that it hardly needs sep- 
arate description. It is more nearly perfect in all its parts than the 
5*, having a larger proportion of its vertical-square diagonals cor- 
rect. Any square group of four, nine, sixteen, twenty-five or thirty- 
six small squares is magic, and if the group of nine or twenty-five 



I 


255 


254 


4 


248 


10 


11 


245 


240 


18 


19 


237 


25 


231 


230 


28 


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6 


7 


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13 


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242 


16 


21 


235 


234 


24 


228 


30 


31 


225 


8 


250 


251 


5 


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15 


14 


244 


233 


23 


22 


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32 


226 


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9 


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239 


17 


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27 


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224 


34 


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221 


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44 


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207 


206 


52 


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58 


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37 


219 


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55 


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61 


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194 


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39 


38 


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223 


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75 


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159 


158 


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106 


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115 


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139 


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127 


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118 


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113 


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123 


122 


136 



Fig. 3- (4 4 ) 

be taken at the center of the figure it is "perfect." The grouping 
of multiples and powers of n is very similar to that already described 
for 5*. 

II. When n is even. 

I. Let «=4, then 8=514. — The numbers may be arranged in 
either of two ways. If we take the diagram for the 4-cube as 



II 



III 



1 


1295 


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9 


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26 


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3 


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128 


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217 


1079 


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219 


1076 


222 


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237 


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239 


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253 


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i°55 


243 


244 


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260 


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1035 


263 


259 


1068 


1067 


231 


232 


230 


1063 


247 


248 


1048 


1047 


1049 


252 


1032 


1031 


267 


268 


266 


1027 


229 


233 


1065 


1066 


1064 


234 


1050 


1046 


250 


249 


251 


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265 


269 


1029 


1030 


1028 


270 


228 


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225 


226 


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105 1 


245 


1054 


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242 


246 


264 


1034 


261 


262 


1037 


>°33 


1075 


218 


220 


1077 


221 


1080 


240 


106 1 


1059 


238 


1058 


235 


1039 


254 


256 


1041 


257 


1044 


865 


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43° 


867 


428 


870 


4H 


884 


885 


4U 


887 


409 


901 


395 


394 


903 


392 


906 


426 


872 


424 


423 


875 


871 


889 


407 


891 


892 


404 


408 


390 


908 


388 


387 


9" 


907 


420 


419 


879 


880 


878 


415 


895 


896 


400 


399 


401 


900 


384 


&S 


915 


916 


914 


379 


877 


881 


417 


418 


416 


882 


402 


398 


898 


897 


899 


397 


9'3 


9'7 


381 


382 


380 


918 


876 


422 


873 


874 


425 


421 


403 


'893 


406 


405 


890 


894 


912 


386 


909 


910 


389 


385 


427 


866 


868 


429 


869 


432 


888 


4'3 


411 


886 


410 


883 


39'- 


902 


904 


393 


9°5 


396 


864 


434 


435 


862 


437 


859 


451 


845 


844 


453 


842 


456 


828 


470 


47' 


826 


473 


823 


439 


857 


441 


442 


854 


858 


840 


458 


838 


837 


461 


457 


475 


821 


477 


478 


818 


822 


445 


446 


850 


849 


8S« 


450 


834 


833 


465 


466 


464 


829 


481 


482 


814 


8>3 


815 


486 


852 


848 


448 


447 


449 


847 


463 


467 


83' 


832 


830 


468 


816 


812 


484 


483 


485 


811 


853 


443 


856 


855 


440 


444 


462 


836 


459 


460 


839 


835 


8i7 


479 


820 


819 


476 


480 


438 


863 


861 


436 


860 


433 


841 


452 


454 


843 


455 


846 


474 


827 


825 


472 


824 


469 


756 


542 


543 


754 


545 


751 


559 


737 


736 


56i 


734 


564 


720 


578 


579 


7'8 


58i 


7'5 


547 


749 


549 


55° 


746 


750 


732 


566 


73° 


729 


569 


565 


583 


7'3 


585 


586 


710 


7'4 


553 


554 


742 


741 


743 


558 


726 


725 


573 


574 


572 


721 


589 


59° 


706 


7°5 


707 


594 


744 


740 


556 


555 


557 


739 


57" 


575 


723 


724 


722 


576 


708 


704 


592 


59' 


593 


703 


745 


55" 


748 


747 


548 


552 


57° 


728 


567 


568 


73" 


727 


709 


587 


712 


7" 


584 


588 


546 


755- 


753 


544 


752 


541 


733 


560 


562 


735 


563 


738 


582 


7W 


7'7 


580 


7J6 


577 



Fig. 4, First Part. (6*: 8=3891) 



IV 



VI 



122$ 


71 


70 


1227 


68 


1230 


1224 


74 


75 


1222 


77 


1219 


120^ 


92 


93 


1204 


95 


1201 


r>r> 


1232 


64 


f>3 


'235 


■23' 


79 


1217 


81 


82 


1214 


1218 


97 


"99 


99 


100 


119C 


1200 


fio 


59 


'239 


1240 


1238 


55 


85 


86 


1210 


1209 


[211 


90 


>°3 


104 


1192 


1191 


"93 


108 


'237 


124! 


57 


58 


56 


1242 


1212 


1208 


88 


87 


89 


1207 


"94 


1190 


106 


■°5 


107 


1189 


1236 


62 


<m 


1234 


65 


61 


1213 


83 


1216 


1215 


80 


84 


"95 


101 


1 198 


"97 


98 


102 


(>7 


1226 


1228 


69 


1229 


72 


78 


1223 


1221 


7* 


1220 


73 


96 


1205 


1203 


94 


1202 


91 


180 


1 1 18 


1 1 19 


178 


1121 


'75 


181 


'"5 


1 114 


'83 


1 1 12 


186 


'99 


1097 


1096 


20 r 


IO94 


204 


1123 


'73 


1125 


I 126 


170 


'74 


11 10 


188 


1108 


1107 


KJI 


187 


1092 


206 


1090 


1089 


209 


205 


1129 


1130 


166 


"•'5 


167 


"34 


1 104 


1103 


'95 


196 


194 


1099 


1086 


1085 


213 


214 


212 


1081 


168 


164 


1132 


"3' 


"33 


""'3 


'93 


'97 


MOI 


1 102 


MOO 


198 


211 


2'5 


1083 


1084 


I082 


216 


169 


1127 


172 


171 


1124 


1128 


192 


1 106 


189 


190 


1 109 


"05 


210 


1088 


207 


208 


I09I 


1087 


1122 


'79 


'77 


1120 


176 


1 1 17 


1111 


182 


184 


"'3 


185 


1 1 16 


1093 


200 


202 


1095 


203 


1098 


1009 


287 


286 


1011 


284 


1014 


ico8 


290 


291 


1006 


293 


1003 


990 


308 


309 


988 


3" 


985 


282 


1016 


280 


279 


1019 


1015 


295 


IOOI 


297 


298 


998 


1002 


3'3 


983 


3'5 


3'6 


980 


984 


276 


275 


1023 


1024 


1022 


271 


30' 


302 


994 


993 


995 


306 


3'9 


320 


976 


975 


977 


324 


102 1 


1025 


273 


274 


272 


1026 


996 


992 


3°4 


303 


305 


99' 


978 


974 


322 


321 


323 


973 


1020 


278 


1017 


1018 


281 


'77 


997 


299 


1000 


999 


296 


300 


979 


3'7 


982 


981 


3'4 


3'8 


283 


1010 


1012 


285 


1013 


288 


294 


1007 


1005 


292 


1004 


289 


312 


989 


987 


3'0 


986 


3°7 


S<" 


935 


934 


363 


932 


366 


360 


938 


939 


358 


94' 


355 


342 


956 


957 


340 


959 


337 


93° 


368 


928 


927 


371 


367 


943 


353 


945 


946 


350 


354 


961 


335 


963 


964 


332 


336 


924 


923 


375 


376 


374 


919 


949 


95° 


346 


345 


347 


954 


967 


968 


328 


327 


329 


972 


373 


377 


921 


922 


920 


378 


348 


344 


952 


95' 


953 


343 


330 


326 


970 


969 


97' 


325 


372 


926 


369 


370 


929 


925 


349 


947 


352 


35' 


944 


948 


33' 


965 


334 


333 


962 


966 


93' 


362 


3 6 4 


933 


365 


936 


942 


359 


357 


940 


356 


937 


960 


34' 


339 


958 


338 


955 


504 


794 


795 


502 


797 


499 


505 


79' 


790 


5°7 


788 


510 


523 


773 


772 


525 


77° 


528 


799 


497 


801 


802 


494 


498 


786 


5'2 


784 


783 


5'5 


5" 


768 


53o 


766 


765 


533 


529 


805 


806 


490 


489 


491 


810 


780 


779 


5'9 


520 


5'8 


775 


762 


76i 


537 


538 


536 


757 


492 


488 


808 


807 


809 


487 


5'7 


52' 


777 


778 


776 


522 


535 


539 


759 


760 


758 


540 


493 


803 


496 


495 


800 


804 


5'6 


782 


5'3 


5'4 


785 


781 


534 


764 


53' 


532 


767 


763 


798 


503 


5°' 


70 


500 


79i 


787 


506 


508 


789 


509 


792 


769 


524 


526 


771 


527 


774 


612 


686 


687 


610 


689 


607 


613 


683 


682 


6'5 


680 


618 


631 


66 S 


664 


633 


662 


636 


691 


605 


693 


694 


602 


606 


678 


620 


676 


675 


623 


619 


660 


638 


658 


657 


641 


637 


697 


698 


598 


597 


599 


702 


672 


671 


627 


628 


626 


667 


654 


653 


645 


646 


644 


649 


£00 


595 


700 


699 


701 


595 


625 


629 


669 


670 


668 


630 


643 


647 


651 


652 


650 


648 


601 


695 


604 


603 


692 


696 


624 


674 


621 


622 


677 


673 


642 


656 


639 


640 


659 


655 


690 


611 


609 


688 


608 


685 


679 


614 


616 


681 


617 


684 


661 


632 


634 


663 


635 


666 



Fig. 4, Second Part. (6':S=389i) 



316 



THE MONIST. 



given in Magic Squares and Cubes and simply extend it to cover 
the larger numbers involved we shall have a group of four cubes 
in which all the "straight" dimensions have S=SI4, but no diag- 
onals except the four cubic diagonals. Each horizontal row of 
squares will produce a cube having exactly the same properties as 
those in the four vertical rows. If the four squares in either diag- 



I 


409S 


4094 


4 


5 


4091 


4090 


8 


4032 


66 


67 


4029 


4028 


70 


71 


4025 


4088 


10 


11 


4085 


4084 


14 


IS 


4081 


73 


4023 


4022 


76 


77 


4019 


4018 


80 


4080 


18 


19 


4077 


4076 


22 


23 


4073 


81 


4015 


4014 


84 


85 


401 1 


4010 


88 


25 


4071 


4070 


28 


29 


4067 


4066 


32 


4008 


90 


91 


4005 


4004 


94 


95 


4001 


406S 


31 


30 


4068 


4069 


27 


26 


4072 


96 


4002 


4003 


93 


92 


4006 


4007 


89 


24 


4074 


4075 


31 


20 


4078 


4079 


17 


4009 


87 


86 


4012 


4013 


83 


82 


4016 


16 


4082 


4083 


13 


12 


4086 


4087 


9 


4017 


79 


78 


4020 


4021 


75 


74 


4024 


4089 


7 


6 


4092 


4093 


3 


2 


4096 


72 


4026 


4027 


69 


68 


4030 


4031 


65 


4064 


34 


35 


4061 


4060 


38 


39 


4057 


97 


3999 


3098 


100 


101 


399S 


3994 


104 


41 


4055 


4054 


44 


45 


4051 


4050 


48 


3992 


106 


107 


3989 


3988 


1 10 


III 


3985 


49 


4047 


4046 


52 


53 


4043 


4042 


56 


3984 


114 


115 


398i 


3980 


118 


119 


3977 


4040 


58 


59 


4037 


4036 


62 


63 


4033 


121 


3975 


3974 


124 


125 


3971 


3970 


128 


64 


4034 


4035 


61 


60 


4038 


4039 


57 


3969 


127 


126 


3972 


3973 


123 


122 


3976 


4041 


55 


54 


4044 


4045 


51 


50 


4048 


120 


3978 


3979 


117 


116 


3982 


3983 


113 


4049 


47 


46 


4052 


4053 


43 


42 


4056 


112 


3986 


3987 


109 


108 


3990 


3991 


ios 


40 


4058 


4059 


37 


36 


4062 


4063 


33 


3993 


103 


ro2 


3996 


3997 


99 


98 


4000 



II IV 

Fig. 5, 8*, First Part (One cube written). 

onal of the figure be piled together neither vertical columns nor 
cubic diagonals will have the correct summation, but all the diagonals 
of vertical squares in either direction will. Regarding the whole 
group of sixteen squares as a plane square we find it magic, having 
the summation 2056 in every rank, file and corner diagonal, 1028 



CRITICISMS AND DISCUSSIONS. 



317 



in each half-rank or half-file, and 514 in each quarter-rank or 
quarter-file. Furthermore all complementary pairs are balanced about 
the center. 

The alternative arrangement shown in Fig. 3 makes each of the 
small squares perfect in itself, with every rank, file and corner diag- 
onal footing up 514 and complementary pairs balanced about the 



3968 


130 


131 


3965 


39&t 


134 


135 


396i 


193 


3903 


3902 


196 


«97 


3899 


3898 


200 


137 


3959 


3958 


140 


141 


3955 


3954 


144 


3896 


202 


203 


3893 


3892 


206 


207 


3889 


145 


3051 


3950 


148 


149 


3947 


3946 


152 


3888 


210 


211 


3885 


3884 


214 


215 


3881 


3944 


154 


155 


3941 


3940 


158 


IS9 


3937 


217 


3879 


3878 


220 


221 


3875 


3874 


224 


160 


3938 


3939 


157 


156 


3942 


3943 


153 


3873 


223 


222 


3876 


3877 


219 


218 


3880 


3945 


151 


150 


3948 


3949 


147 


146 


3952 


216 


38R? 


3883 


213 


212 


3886 


3887 


209 


3953 


143 


142 


3956 


3957 


139 


138 


3960 


208 


3890 


3891 


205 


204 


3894 


3895 


201 


136 


3962 


3963 


133 


132 


3966 


3967 


129 


3897 


199 


198 


3900 


3901 


195 


194 


3904 


161 


3935 


3934 


164 


165 


3931 


393° 


168 


3872 


226 


227 


3869 


3868 


230 


231 


3865 


3928 


170 


171 


3925 


3924 


»74 


175 


3921 


233 


3863 


3862 


236 


237 


3859 


38S8 


240 


3920 


178 


179 


35W7 


39i6 


182 


183 


3913 


241 


3855 


3854 


244 


245 


3851 


3850 


248 


185 


39" 


39«o 


188 


«89 


3907 


3906 


192 


.1848 


250 


251 


3845 


3844 


254 


255 


3841 


3905 


191 


190 


3908 


3909 


187 


186 


3912 


256 


3842 


3843 


253 


252 


3846 


3847 


249 


184 


3914 


3915 


181 


180 


3918 


3919 


177 


3849 


247 


246 


3852 


3853 


243 


242 


3856 


176 


3922 


3923 


173 


172 


3926 


3927 


169 


3857 


239 


238 


3860 


3861 


335 


234 


3864 


3929 


167 


166 


3932 


3933 


163 


162 


3936 


232 


3866 


3867 


229 


228 


3870 


3871 


225 



VI VIII 

Fig. 5, 8*, Second Part (One cube written). 



center. As in the other arrangement the squares in each vertical 
or horizontal row combine to make cubes whose "straight" dimen- 
sions all have the right summation. In addition the new form has 
the two plane diagonals of each original square (eight for each 
cube), but sacrifices the four cubic diagonals in each cube. In lieu 



318 THE MONIST. 

of these we find a complete set of "bent diagonals" ("Franklin") 
like those described for the magic cube of six in The Monist for 
July, 1909. 

If the four squares in either diagonal of the large figure be 
piled up it will be found that neither cubic diagonal nor vertical 
column is correct, but that all diagonals of vertical squares facing 
toward front or back are. Taken as a plane figure the whole group 
makes up a magic square of 16 with the summation 2056 in every 
rank, file or corner diagonal, half that summation in half of each 
of those dimensions, and one-fourth of it in each quarter dimension. 

2. Let n=6, then 8=3891. — With the natural series 1. . .1296 
squares were constructed which combined to produce the six magic 
cubes of six indicated by the Roman numerals in Fig. 4. These 
have all the characteristics of the 6-cube described in The Monist 
of July last — 108 "straight" rows, 12 plane diagonals and 24 "bent" 
diagonals in each cube, with the addition of 32 vertical-square 
diagonals if the squares are piled in a certain order. A seventh 
cube with the same features is made by combining the squares in 
the lowest horizontal row — i. e., the bottom squares of the num- 
bered cubes. The feature of the cubic bent diagonals is found on 
combining any three of the small squares, no matter in what order 
they are taken. In view of the recent discussion of this cube it seems 
unnecessary to give any further account of it now. 

The whole figure, made up as it is of thirty-six magic squares, 
is itself a magic square of 36 with the proper summation (23346) 
for every rank, file and corner diagonal, and the corresponding 
fractional part of that for each half, third or sixth of those dimen- 
sions. Any square group of four, nine, sixteen or twenty-five of the 
small squares will be magic in all its dimensions. 

3. Let n=8, then 8=16388. — The numbers 1...4096 may be 
arranged in several different ways. If the diagrams in Mr. An- 
drews's book be adopted we have a group of eight cubes in which 
rank, file, column and cubic diagonal are correct (and in which 
the halves of these dimensions have the half summation), but all 
plane diagonals are irregular. If the plan be adopted of construct- 
ing the small squares of complementary couplets, as in the 6-cube, 
the plane diagonals are equalized at the cost of certain other features. 
I have used therefore a plan which combines to some extent the ad- 
vantages of both the others. 

It will be noticed that each of the small squares in Fig. 5 is 



CRITICISMS AND DISCUSSIONS. 319 

perfect in that it has the summation 16388 for rank, file and corner 
diagonal (also for broken diagonals if each of the separated parts 
contain two, four or six — not an odd number of cells) , and in balan- 
cing complementary couplets. When the eight squares are piled 
one upon the other a cube results in which rank, file, column, the 
plane diagonals of each horizontal square, the four ordinary cubic 
diagonals and 32 cubic bent diagonals all have S=i6388. What is 
still more remarkable, the half of each of the "straight" dimensions 
and of each cubic diagonal has half that sum. Indeed this cube of 
eight can be sliced into eight cubes of 4 in each of which every rank, 
file, column and cubic diagonal has the footing 8194; and each of 
these 4-cubes can be subdivided into eight tiny 2-cubes in each of 
which the eight numbers foot up 16388. 

So much for the features of the single cube here presented. 
As a matter of fact only the one cube has actually been written out. 
The plan of its construction, however, is so simple and the relations 
of numbers so uniform in the powers of 8 that it was easy to in- 
vestigate the properties of the whole 8 4 scheme without having the 
squares actually before me. I give here the initial number of each 
of the eight squares in each of the eight cubes, leaving it for some 
one possessed of more leisure to write them all out and verify my 
statements as to the intercubical features. It should be remembered 
that in each square the number diagonally opposite the one here 
given is its complement, i. e., the number which added to it will 
give the sum 4097. 



I 


II 


III 


IV 


V 


VI 


VII 


VIII 


I 


3840 


3584 


769 


3072 


1281 


1537 


2304 


4064 


289 


545 


3296 


1057 


2784 


2528 


1825 


4032 


321 


577 


3264 


1089 


2752 


2496 


1857 


97 


3744 


3488 


865 


2976 


1377 


1633 


2208 


3968 


385 


641 


3200 


1 153 


2688 


2432 


1921 


161 


3680 


3424 


929 


2912 


1441 


1697 


2144 


193 


3648 


3392 


961 


2880 


1473 


1729 


2112 


3872 


481 


737 


3104 


1249 


2592 


2336 


2017 



16388 16388 16388 16388 16388 16388 16388 16388 

Each of the sixty-four numbers given above will be at the 
upper left-hand corner of a square and its complement at the lower 
right-hand corner. The footings given are for these initial numbers, 



320 THE MONIST. 

but the arrangement of numbers in the squares is such that the 
footing will be the same for every one of the sixty-four columns 
in each cube. If the numbers in each horizontal line of the table 
above be added they will be found to have the same sum: conse- 
quently the squares headed by them must make a cube as nearly 
perfect as the example given in Fig. 5, which is cube I of the table 
above. But the sum of half the numbers in each line is half of 
16388, and hence each of the eight cubes formed by taking the 
squares in the horizontal rows is capable of subdivision into 4-cubes 
and 2-cubes, like our original cube. We thus have sixteen cubes, 
each with the characteristics described for the one presented in 

Fig- 5- 

If we pile the squares lying in the diagonal of our great square 
(starting with 1, 289, etc., or 2304, 2528, etc.) we find that its col- 
umns and cubic diagonals are not correct; but all the diagonals of 
its vertical squares are so, and even here the remarkable feature of 
the half-dimension persists. 

Of course there is nothing to prevent one's going still further 
and examining constructions involving the fifth or even higher pow- 
ers, but the utility of such research may well be doubted. The purpose 
of this article is to suggest in sketch rather than to discuss exhaus- 
tively an interesting field of study for some one who may have time 
to develop it. 

H. M. Kingery. 

Wabash College.